TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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Alexandrov's theorem, under Dependent Choice: a subspace of a complete metric space is completely metrizable exactly when it is $G_{δ}$

Statement

Assume Dependent Choice. For a subspace $Y$ of a complete metric space $X$ , $Y$ is completely metrizable if and only if $Y$ is a $G_{δ}$ subset of $X$ .

Facts & Assumptions

Given: The objects, hypotheses, and choice principles stated above.

[F1]

If $(X, d)$ is a complete metric space and $Y \subseteq X$ is $G_{δ}$ in $X$ , then the subspace $Y$ is completely metrizable. (Under the Axiom of Countable Choice, every $G_{δ}$ subspace of a complete metric space is completely metrizable).

[F2]

Assume Dependent Choice. If $Y$ is a completely metrizable subspace of a metric space $X$ , then $Y$ is a $G_{δ}$ subset of $X$ . (Under Dependent Choice, every completely metrizable subspace of a metric space is $G_{δ}$ ).

Proof

technique · direct

1.1

The empty subspace satisfies both conditions.

givenF1F2

2.1

For a nonempty subspace apply the two preceding implications with the induced topology.

step 1.1F1F2

3.1

In the reverse direction use the given complete ambient metric; in the forward direction use only the existence of a compatible complete metric on the subspace, not completeness of the inherited metric.

step 2.1F1F2

4.1

The preceding construction and implications establish the assertion.

step 3.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 56 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

David Marker, Descriptive Set Theory, §§1–2 (standard reference, not scraped)
Michael Kunzinger, General Topology, §§11.3–11.4 (standard reference, not scraped)
MFF General Topology course summary, §4.3 (standard reference, not scraped)
Jesse Peterson, Real Analysis, §§3.6–3.7 (standard reference, not scraped)

Alexandrov's theorem, under Dependent Choice: a subspace of a complete metric space is completely metrizable exactly when it is GδGδ​